Use the given derivative inside parts

$\frac d{dx}e^{x^2}=2xe^{x^2}$. Reversing this gives $\int2xe^{x^2}\,dx=e^{x^2}+C^{\prime}$.

Rewrite $\int x^3e^{x^2}\,dx=\frac12\int x^2(2xe^{x^2})\,dx$. Choose $u=x^2$, $du/dx=2x$, $dv/dx=2xe^{x^2}$ and use (i) to take $v=e^{x^2}$. Hence the result is $\frac12[x^2e^{x^2}-\int2xe^{x^2}\,dx]=\frac12(x^2e^{x^2}-e^{x^2})+C$.